Skew-Product Dynamical Systems

Abstract

This chapter discusses the skew-product dynamical systems. The skew-product flow is the appropriate setting for studying many of the qualitative properties of nonautonomous ordinary differential equations, functional differential equations, finite difference equations, and mappings of manifolds. Its appropriateness derives from the fact that many important questions can be treated within the framework of the dynamics of compact spaces while retaining a setting in which one's geometric intuition can freely play a role. For functional differential equations of retarded type, one considers the product space C X Y, where C is the Banach space of continuous functions from [−r, 0] to Rn with the sup norm and Y a space of functional ƒ: R × C → Rn. Even when ƒ satisfies conditions guaranteeing uniqueness and continuation to the positive reals of a bounded solution, it may still be the case that such a solution cannot be extended to the negative reals.